A generalized Rayleigh–Taylor condition for the Muskat problem

Abstract

In this paper we consider the evolution of two fluid phases in a porous medium. The fluids are separated from each other and also the wetting phase from air by interfaces which evolve in time. We reduce the problem to an abstract evolution equation. A generalised Rayleigh-Taylor condition characterizes the parabolicity regime of the problem and allows us to establish a general well-posedness result and to study stability properties of flat steady-states. When considering surface tension effects at the interface between the fluids and if the more dense fluid lies above, we find bifurcating finger-shaped equilibria which are all unstable.2010 Mathematics Subject Classification. 35B35; 35B36; 35K55; 35R37.

Existence and stability of solutions for a strongly coupled system modelling thin fluid films

“…a < b, provided the density jump across the interface is small compared to A−B. In the case when L = π relation (3.2) is equivalent to γ + g(ρ − − ρ + ) > 0, which is exactly the condition found for the full Muskat problem studied in [7]. The critical case of stability when neither (3.2) nor (3.5) are satisfied will be discussed in Theorem 3.6.…”

“…This is exactly the situation which is encountered when considering the full problem, cf. [7]. However, in contrast to [7], where the nontrivial steady-state solutions have been obtained in the same regime (3.5) but via bifurcation, in the setting considered herein they are solutions of a linear system of ordinary differential equations with constant coefficients and form linear one-dimensional invariant subspaces of the phase space, i.e.…”

“…In contrast, in the present paper we derive the needed a priori estimates by introducing weighted scalar products which dependent on the point where we linearize and by using the Lax-Milgram theorem. Compared to the full Muskat problem [7], where the linearization is a 2 × 2-matrix generator whose off-diagonal entries are of lower order, herein we obtain a matrix with elements having all the same order. Additionally, we identify the steady-state solutions of the problem and study their stability properties.…”

“…The steady states are either flat, or when the fluid below is less dense then that above, there may exist non-constant steady-state solutions of (1.1). They form one-dimensional linear center manifolds which may be regarded as the first order expansions of the analytic bifurcation branches consisting of non-flat steady-state solutions of the Muskat problem in [7].…”

“…The system (1.1a) is supplemented by initial conditions for f and h of the form The model has been obtained in [6] by passing to the limit of thin fluid threads in the full two-phase Muskat problem considered in [7]. It is worth noticing that similar methods have been used in [9] and [14] to find, in the limit of thin fluid threads, thin film equations as approximations for the capillary drive Stokes and Hele-Shaw moving boundary value problems.…”